Problem: Simplify and expand the following expression: $ \dfrac{4}{x + 10}+ \dfrac{4}{2x + 8}- \dfrac{4}{x^2 + 14x + 40} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the second term: $ \dfrac{4}{2x + 8} = \dfrac{4}{2(x + 4)}$ We can factor the quadratic in the third term: $ \dfrac{4}{x^2 + 14x + 40} = \dfrac{4}{(x + 10)(x + 4)}$ Now we have: $ \dfrac{4}{x + 10}+ \dfrac{4}{2(x + 4)}- \dfrac{4}{(x + 10)(x + 4)} $ The least common multiple of the denominators is: $ (x + 10)(x + 4)$ In order to get the first term over $(x + 10)(x + 4)$ , multiply by $\dfrac{2(x + 4)}{2(x + 4)}$ $ \dfrac{4}{x + 10} \times \dfrac{2(x + 4)}{2(x + 4)} = \dfrac{8(x + 4)}{(x + 10)(x + 4)} $ In order to get the second term over $(x + 10)(x + 4)$ , multiply by $\dfrac{x + 10}{x + 10}$ $ \dfrac{4}{2(x + 4)} \times \dfrac{x + 10}{x + 10} = \dfrac{4(x + 10)}{(x + 10)(x + 4)} $ In order to get the third term over $(x + 10)(x + 4)$ , multiply by $\dfrac{2}{2}$ $ \dfrac{4}{(x + 10)(x + 4)} \times \dfrac{2}{2} = \dfrac{8}{(x + 10)(x + 4)} $ Now we have: $ \dfrac{8(x + 4)}{(x + 10)(x + 4)} + \dfrac{4(x + 10)}{(x + 10)(x + 4)} - \dfrac{8}{(x + 10)(x + 4)} $ $ = \dfrac{ 8(x + 4) + 4(x + 10) - 8} {(x + 10)(x + 4)} $ Expand: $ = \dfrac{8x + 32 + 4x + 40 - 8}{2x^2 + 28x + 80} $ $ = \dfrac{12x + 64}{2x^2 + 28x + 80}$ Simplify: $ = \dfrac{6x + 32}{x^2 + 14x + 40}$